7-7 Geometric Sequences as Exponential Functions Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1. 200, 40, 8, … SOLUTION: Since the ratios are constant, the sequence is geometric. The common ratio is . 2. 2, 4, 16, … SOLUTION: The ratios are not constant, so the sequence is not geometric. There is no common difference, so the sequence is not arithmetic. Thus, the sequence is neither geometric nor arithmetic. 3. −6, −3, 0, 3, … SOLUTION: The ratios are not constant, so the sequence is not geometric. Since the differences are constant, the sequence is arithmetic. The common difference is 3. 4. 1, −1, 1, −1, … SOLUTION: Since the ratios are constant, the sequence is geometric. The common ratio is –1. Find the next three terms in each geometric sequence. 5. 10, 20, 40, 80, … SOLUTION: eSolutions Manual - Powered by Cognero Page 1 The common ratio is 2. Multiply each term by the common ratio to find the next three terms. 80 × 2 = 160 160 × 2 = 320 320 × 2 = 640 The next three terms of the sequence are 160, 320, and 640. 6. 100, 50, 25, … SOLUTION: Calculate common ratio. The common ratio is 0.5. Multiply each term by the common ratio to find the next three terms. 25 × 0.5 = 12.5 12.5 × 0.5 = 6.25 6.25 × 0.5 = 3.125 The next three terms of the sequence are 12.5, 6.25, and 3.125. 7. 4, −1, , … SOLUTION: Calculate the common ratio. The common ratio is . Multiply each term by the common ratio to find the next three terms. × = × = × = The next three terms of the sequence are , , and . 8. −7, 21, −63, … SOLUTION: Calculate the common ratio. The common ratio is –3. Multiply each term by the common ratio to find the next three terms. –63 × –3 = 189 189 × –3 = –567 –567 × –3 = 1701 The next three terms of the sequence are 189, −567, and 1701. Write an equation for the nth term of the geometric sequence, and find the indicated term. 9. Find the fifth term of −6, −24, −96, … SOLUTION: Calculate the common ratio. n – 1 Use the formula an = a1r to write an equation for the nth term of the geometric series. The common ratio is 4, n−1 so r = 4. The first term is –6, so a1 = –6. Then, an = −6 • (4) . The 5th term of the sequence is –1536. 10. Find the seventh term of −1, 5, −25, … SOLUTION: Calculate the common ratio. n – 1 Use the formula an = a1r to write an equation for the nth term of the geometric series. The common ratio is –5, n−1 so r = –5. The first term is –1, so a1 = –1. Then, an = −1 • (–5) . The 7th term of the sequence is –15,625. 11. Find the tenth term of 72, 48, 32, … SOLUTION: Calculate the common ratio. n – 1 Use the formula an = a1r to write an equation for the nth term of the geometric series. The common ratio is , so r = . The first term is 72, so a1 = 72. Then, an = 72 • . The 10th term of the sequence is . 12. Find the ninth term of 112, 84, 63, … SOLUTION: Calculate the common ratio. n – 1 Use the formula an = a1r to write an equation for the nth term of the geometric series. The common ratio is , so r = . The first term is 112, so a1 = 112. Then, an = 112 • . The 9th term of the sequence is . 13. EXPERIMENT In a physics class experiment, Diana drops a ball from a height of 16 feet. Each bounce has 70% the height of the previous bounce. Draw a graph to represent the height of the ball after each bounce. SOLUTION: Make a table of values. Bounce Ball Height 1 0.7(16) = 11.2 2 0.7(11.2) = 7.84 3 0.7(7.84) = 5.488 4 0.7(5.488) = 3.8416 5 0.7(3.8416) = 2.68912 6 0.7(2.68912) = 1.882384 7 0.7(1.882384) =1.3176688 Graph the bounce on the x-axis and the ball height on the y-axis. Determine whether each sequence is arithmetic, geometric, or neither. Explain. 14. 4, 1, 2, … SOLUTION: Find the ratios of consecutive terms. The ratios are not constant, so the sequence is not geometric. Find the ratios of the differences of consecutive terms There is no common difference, so the sequence is not arithmetic. Thus, the sequence is neither geometric nor arithmetic. 15. 10, 20, 30, 40 … SOLUTION: Find the ratios of consecutive terms. The ratios are not constant, so the sequence is not geometric. Find the differences of consecutive terms. Since the differences are constant, the sequence is arithmetic. The common difference is 10. 16. 4, 20, 100, … SOLUTION: Find the ratios of consecutive terms. Since the ratios are constant, the sequence is geometric. The common ratio is 5. 17. 212, 106, 53, … SOLUTION: Find the ratios of consecutive terms. Since the ratios are constant, the sequence is geometric. The common ratio is . 18. −10, −8, −6, −4 … SOLUTION: Find the ratios of consecutive terms. The ratios are not constant, so the sequence is not geometric. Find the differences of consecutive terms. Since the differences are constant, the sequence is arithmetic. The common difference is 2. 19. 5, −10, 20, 40, … SOLUTION: Find the ratios of consecutive terms. The ratios are not constant, so the sequence is not geometric. Find the differences of consecutive terms. There is no common difference, so the sequence is not arithmetic. Thus, the sequence is neither geometric nor arithmetic. Find the next three terms in each geometric sequence. 20. 2, −10, 50, … SOLUTION: Calculate the common ratio. The common ratio is –5. Multiply each term by the common ratio to find the next three terms. 50 × –5 = –250 –250 × –5 = 1250 1250 × –5 = –6250 The next three terms of the sequence are −250, 1250, and −6250. 21. 36, 12, 4, … SOLUTION: Calculate the common ratio. The common ratio is . Multiply each term by the common ratio to find the next three terms. 4 × = × = × = The next three terms of the sequence are , , and . 22. 4, 12, 36, … SOLUTION: Calculate the common ratio. The common ratio is 3. Multiply each term by the common ratio to find the next three terms. 36 × 3 = 108 108 × 3 = 324 324 × 3 = 972 The next three terms of the sequence are 108, 324, and 972. 23. 400, 100, 25, … SOLUTION: Calculate the common ratio. The common ratio is . Multiply each term by the common ratio to find the next three terms. 25 × = × = × = The next three terms of the sequence are , , and . 24. −6, −42, −294, … SOLUTION: Calculate the common ratio. The common ratio is 7. Multiply each term by the common ratio to find the next three terms. –294 × 7 = –2058 –2058 × 7 = –14,406 –14,406 × 7 = –100,842 The next three terms of the sequence are −2058, −14,406, and −100,842. 25. 1024, −128, 16, … SOLUTION: Calculate the common ratio. The common ratio is . Multiply each term by the common ratio to find the next three terms. 16 × = –2 –2 × = × = The next three terms of the sequence are −2, , and . 26. The first term of a geometric series is 1 and the common ratio is 9. What is the 8th term of the sequence? SOLUTION: The 8th term of the sequence is 4,782,969. 27. The first term of a geometric series is 2 and the common ratio is 4. What is the 14th term of the sequence? SOLUTION: The 14th term of the sequence is 134,217,728. 28. What is the 15th term of the geometric sequence −9, 27, −81, …? SOLUTION: Calculate the common ratio. The common ratio is –3. The 15th term of the sequence is –43,046,721. 29. What is the 10th term of the geometric sequence 6, −24, 96, …? SOLUTION: Calculate the common ratio. The common ratio is –4. The 10th term of the sequence is –1,572,864. 30. PENDULUM A pendulum swings with an arc length of 24 feet on its first swing. On each swing after the first swing, the arc length is 60% of the length of the previous swing. Draw a graph that represents the arc length after each swing. SOLUTION: Make a table of values. Swing Arc Length 1 24 2 0.6(24) = 14.4 3 0.6(14.4) = 8.64 4 0.6(8.64) = 5.184 5 0.6(5.184) = 3.1104 6 0.6(3.1104) = 1.86624 Graph the swing on the x-axis and the arc length on the y-axis. 31. Find the eighth term of a geometric sequence for which a3 = 81 and r = 3. SOLUTION: Because a3 = 81, the third term in the sequence is 81. To find the eighth term of the sequence, you need to find the 1st term of the sequence. Use the nth term of a Geometric Sequence formula. Then a1 is 9. Use a1 to find the eighth term of the sequence. The eighth term of the geometric sequence is 19,683. 32. CCSS REASONING At an online mapping site, Mr. Mosley notices that when he clicks a spot on the map, the map zooms in on that spot. The magnification increases by 20% each time. a. Write a formula for the nth term of the geometric sequence that represents the magnification of each zoom level. (Hint: The common ratio is not just 0.2.) b. What is the fourth term of this sequence? What does it represent? SOLUTION: a.